Page 88 - Glucose Monitoring Devices

The state-of-the-art modeling method by Vettoretti et al. 87

When V ¼ 2, f YjV (yj2) is described by an exponential PDF model shifted right of
T 2 :
(
l 2 e l 2 ðy T 2 Þ y T 2
f YjV ðyj2Þ¼ (5.6)
0 y < T 2
Parameters l 1 and l 2 are estimated by ML by inverting the sample mean of left or
right outliers properly shifted and reversed. Finally, the PDF of Y is obtained as
follows:

f Y ðyÞ¼ f YjV ðyj0Þ$p V ð0Þþ f YjV ðyj1Þ$p V ð1Þþ f YjV ðyj2Þ$p V ð2Þ (5.7)
where the values p V (1) and p V (2) are estimated by dividing the number of observed
left or right outliers by the total number of observations in the specific constant-SD
zone, while p V (0) is found as follows:
p V ð0Þ¼ 1 p V ð1Þ p V ð2Þ (5.8)
Remark. We propose for step C the use of canonical PDF models, like the skew-
normal and the exponential, as they can be easily used to generate random samples
in in silico applications. Conversely, the use of a noncanonical model, for example, a
spline model, would require much more complex procedures to generate random
samples from the model.

Model validation
The model identified in each constant-SD zone for absolute or relative error can be
validated by comparing the distribution of random samples simulated by the model
with the distribution of the test set data. Model validation is an important step to
perform before using the SMBG error model for simulation applications because
it allows to check if the model is able to generate realistic realizations of the
SMBG error. Such a validation can be performed by calculating the mean absolute
difference (MAD) between the empirical distribution function of random samples
simulated by the model and the empirical distribution function of the test set data,
and by the application of goodness-of-fit tests, like the two-sample Kolmogorove
Smirnov (KS) and Crame ´revon Mises (CvM) tests.
For such a purpose, M random samples are simulated by drawing n test values
from the model identified in each zone, being n test the cardinality of the test set.
In particular, SMBG error values are drawn from the exponential PDF of left outliers
with probability p V (1), from the exponential PDF of right outliers with probability
p V (2) and from the skew-normal PDF of nonoutliers with probability p V (0). If the
model does not include outliers’ description, then p V (1) ¼ p V (2) ¼ 0 and all the n test
SMBG error values are drawn from the skew-normal PDF. To sample a random num-
ber y from a skew-normal PDF, with generic parameters x, u, and a, a three-step
method is provided in Ref. [43]. First, two values u 0 and u 1 having marginal PDF
.
p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
N(0,1) and correlation d ¼ a 1 þ a are generated by sampling u 0 and r from